Globally Convergent Second-order Schemes for Minimizing Twice-differentiable Functions

In this paper, we suggest new universal second-order methods for unconstrained minimization of twice-differentiable (convex or non-convex) objective function. For the current function, these methods automatically achieve the best possible global complexity estimates among different H older classes containing the Hessian of the objective. The universal methods for functional residual and for norm of the gradient are different. For development of the latter methods, we introduced a new line-search acceptance criterion, which can be seen as a nonlinear modification of the Armijo-Goldstein condition.